PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Null-homotopic maps form a two-sided additive ideal

Statement

If $A$ is an abelian category, then in $Ch (A)$ the null-homotopic maps form a two-sided additive ideal: the zero map is null-homotopic, sums of null-homotopic maps are null-homotopic, and whiskering a null-homotopic map on either side by a chain map again gives a null-homotopic map.

Facts & Assumptions

Given: An abelian category $A$ , null-homotopic chain maps $f, g : C_{∙} \to D_{∙}$ , and chain maps $v : B_{∙} \to C_{∙}$ , $u : D_{∙} \to E_{∙}$ in $Ch (A)$ .

[L1]

A null-homotopic chain map is a chain map homotopic to the zero map (A null-homotopic chain map).

[L2]

Chain homotopy is compatible with sums and whiskering (Chain homotopy is compatible with addition and composition).

[L3]

Because $A$ is abelian and hence additive, $Ch (A)$ is additive, so it has zero maps and sums of parallel maps (The category of complexes in an additive category is additive).

Proof

technique · direct

1.1

The zero chain map is null-homotopic via the zero degree- $1$ family, and [L3] guarantees this zero map exists in $Ch (A)$ .

L1L3givenalgebra

2.1

Because $f$ and $g$ are each homotopic to zero by [L1], [L2] shows $f + g ≃ 0 + 0 = 0$ , $u f ≃ 0$ , and $f v ≃ 0$ . Hence sums and left or right composition preserve null-homotopy, so these maps form a two-sided additive ideal.

L1L2step 1.1algebra∎

Depends on

Used by

Homotopy classes of chain maps Definition
Composition of homotopy classes is well defined Lemma
The homotopy category is additive Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 13.8: The homotopy category (standard reference, not scraped)